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arXiv:1611.07664 · Physical Review A

Scaling of geometric phase and fidelity susceptibility across the critical points and their relations

Jia-Ming Cheng, Ming Gong, Guang‐Can Guo, Zheng-Wei Zhou

Published Jun 21, 2017• 8 citations
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Abstract

It has been found via numerical simulations that the geometric phase (GP) and fidelity susceptibility (FS) across the quantum critical points exhibit some universal scaling laws. Here we propose a singular function expansion method to find their exact singular forms and the related coefficients across the critical points. For models where the gaps are closed and reopened at special points (${k}_{0}=0,\ensuremath{\pi}$), scaling laws can be found as a function of the system length $N$ and parameter deviation $\ensuremath{\lambda}\ensuremath{-}{\ensuremath{\lambda}}_{c}$, where ${\ensuremath{\lambda}}_{c}$ refers to one of the critical parameters. Although the GP and FS are defined in totally different ways, we find that these two measurements are essentially determined by the same physics, and as a consequence, their coefficients are closely related. Some of these exact relations are found in the anisotropic XY model and extended Ising models. We also show that the constant term in FS may be accompanied by a discontinuous jump across the critical points and, thus, does not have a universal scaling form. These findings should be in contrast to the cases where the gaps are not closed and reopened at the special points, in which some of the above scaling laws may break down as a function of the system length. Finally, we investigate the second-order derivative of GP, which may also exhibit some scaling laws across the critical point. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions and may even find important applications in related physical quantities, such as entanglement, discord, correlation, and quantum Euler numbers, which may also exhibit scaling laws across the critical points.

DOI
10.1103/physreva.95.062117

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It has been found via numerical simulations that the geometric phase (GP) and fidelity susceptibility (FS) across the quantum critical points exhibit some universal scaling laws. Here we propose a singular function expansion method to find their exact singular forms and the related coefficients across the critical points. For models where the gaps are closed and reopened at special points (k0=0,\ensuremathπ{k}_{0}=0,\ensuremath{\pi}k0​=0,\ensuremathπ), scaling laws can be found as a function of the system length NNN and parameter deviation \ensuremathλ\ensuremath−\ensuremathλc\ensuremath{\lambda}\ensuremath{-}{\ensuremath{\lambda}}_{c}\ensuremathλ\ensuremath−\ensuremathλc​, where \ensuremathλc{\ensuremath{\lambda}}_{c}\ensuremathλc​ refers to one of the critical parameters. Although the GP and FS are defined in totally different ways, we find that these two measurements are essentially determined by the same physics, and as a consequence, their coefficients are closely related. Some of these exact relations are found in the anisotropic XY model and extended Ising models. We also show that the constant term in FS may be accompanied by a discontinuous jump across the critical points and, thus, does not have a universal scaling form. These findings should be in contrast to the cases where the gaps are not closed and reopened at the special points, in which some of the above scaling laws may break down as a function of the system length. Finally, we investigate the second-order derivative of GP, which may also exhibit some scaling laws across the critical point. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions and may even find important applications in related physical quantities, such as entanglement, discord, correlation, and quantum Euler numbers, which may also exhibit scaling laws across the critical points.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse XY\mathit{XY}XY model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size LLL is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.
Within the numerically exact solution to the Dicke model proposed previously, we study the quantum criticality in terms of the ground-state (GS) energy, fidelity, and the order parameter. The finite size scaling analysis for the average fidelity susceptibility (FS) and second derivative of GS energy are performed. The correlation length exponent is obtained to be $\ensuremath{\nu}=2/3$, which is the same as that in Lipkin-Meshkov-Glick model obtained previously, suggesting the same universality. It is observed that average FS and second derivative of GS energy show similar critical behavior, demonstrating the intrinsic relation in the Dicke model. The scaling behavior for the order parameter and the singular part of the GS energy at the critical point are also analyzed and the obtained exponents are consistent with the previous scaling hypothesis in 1/N1/N1/N expansion scheme.
The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity susceptibility, and the correlation length extracted from the curvature function also gives a momentum scale over which the fidelity susceptibility decays. To map out the profile and criticality of the fidelity susceptibility, we turn to quantum walks that simulate one-dimensional class BDI and two- dimensional class D Dirac models and demonstrate their accuracy in capturing the critical exponents and scaling laws near topological phase transitions.
We study the scaling behavior of fidelity susceptibility density $({\ensuremath{\chi}}_{\mathrm{f}})$ at or close to an anisotropic quantum critical point (AQCP) characterized by two different correlation length exponents ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an AQCP is different from that seen near an isotropic QCP. In particular, for a finite system with linear dimension L∣∣{L}_{||}L∣∣​ (${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with L∣∣{L}_{||}L∣∣​ for small L∣∣{L}_{||}L∣∣​, with an exponent depending on both ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and eventually crosses over to a scaling with ${L}_{\ensuremath{\perp}}$ for ${L}_{||}^{1/{\ensuremath{\nu}}_{||}}\ensuremath{\gtrsim}{L}_{\ensuremath{\perp}}^{1/{\ensuremath{\nu}}_{\ensuremath{\perp}}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an AQCP and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.
We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.